Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits

We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by $(\beta ,\theta )$ - the “inverse temperature” and the “chemical potential”. We prove that the locally averaged spin field rescales to the solution of the dynamical ${\mathrm{\Phi }}^4$ equation near a curve in the $(\beta ,\theta )$ plane and to the solution of the dynamical ${\mathrm{\Phi }}^6$ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.